Random pro-p groups, braid groups, and random tame Galois groups

The paper proposes a kind of “non-abelian Cohen-Lenstra heuristic.” A typical prediction: if S is a randomly chosen pair of primes, each of which is congruent to 5 mod 8, and G_S(p) is the Galois group of the maximal pro-2 extension of Q unramified away from S, then G_S(p) is infinite 1/16 of the time.

The usual Cohen-Lenstra conjectures — well, there are a lot of them, but the simplest one asks: given an odd prime p and a finite abelian p-group A, what is the probability P(A) that a randomly chosen quadratic imaginary field K has a class group whose p-primary part is isomorphic to A? (Note that the existence of P(A) — which we take to be a limit in X of the corresponding probability as K ranges over quadratic imaginary fields of discriminant at most X — is not at all obvious, and in fact is not known for any p!)

Cohen and Lenstra offered a beautiful conjectural answer to that question: they suggested that the p-parts of class groups were uniformly distributed among finite abelian p-groups. And remember — that means that P(A) should be proportional to 1/|Aut(A)|. (See the end of this post for more on uniform distribution in this categorical setting.)

Later, Friedman and Washington observed that the Cohen-Lenstra conjectures could be arrived at by another means: if you take K to be the function field of a random hyperelliptic curve X over a finite field instead of a random quadratic imaginary field, then the finite abelian p-group you’re after is just the cokernel of F-1, where F is the matrix corresponding to the action of Frobenius on T_p Jac(X). If you take the view that F should be a “random” matrix, then you are led to the following question:

Let F be a random element of GL_N(Z_p) in Haar measure: what is the probability that coker(F-1) is isomorphic to A?

And this probability, it turns out, is precisely the P(A) conjectured by Cohen-Lenstra.

(But now you cry out: but Frobenius isn’t just any old matrix! It’s in the generalized symplectic group! Yes — and Jeff Achter has shown that, at least as far as the probability distribution on A/pA goes, the “right” random matrix model gives you the same answer as the Friedman-Washington quick and dirty model. Phew.)

Now, in place of a random quadratic imaginary field, pick a prime p and a random set S of g primes, each of which is 1 mod p. As above, let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S; this is a pro-p group of rank g. What can we say about the probability distribution on G_S(p)? That is, if G is some pro-p group, can we compute the probability that G_S(p) is isomorphic to G?

Again, there are two approaches. We could ask that G_S(p) be a “random pro-p group of rank g.” But this isn’t quite right; G_S(p) has extra structure, imparted to it by the images in G_S(p) of tame inertia at the primes of S. We define a notion of “pro-p group with inertia data,” and for each pro-p GWID G we guess that the probability that G_S(p) = G is proportional to 1/Aut(G); where Aut(G) refers to the automorphisms of G as GWID, of course.

On the other hand, you could ask what would happen in the function field case if the action of Frobenius on — well, not the Tate module of the Jacobian anymore, but the full pro-p geometric fundamental group of the curve — is “as random as possible.” (In this case, the group from which Frobenius is drawn isn’t a p-adic symplectic group but Ihara’s “pro-p braid group.”)

And the happy conclusion is that, just as in the Cohen-Lenstra setting, these two heuristic arguments yield the same prediction. For the relatively few pro-p groups G such that we can compute Pr(G_S(p) = G), our heuristic gives the right answer. For several more, it gives an answer that seems consistent with numerical experiments.

Maybe you’ve thought about the following question: is there a good notion of “random” Galois representation of G_Q into GLn(F_q) (say for n and q fixed)? One could of course ask the same for geometric l-adic Galois representations, which is perhaps more interesting. This problem seems connected to yours, since the kernels of these rep’ns will give interesting Galois extensions with Chevalley groups as their Galois groups.

[…] Emmanuel pointed me to a very interesting recent paper by Kenneth Maples, a grad student at UCLA working under Terry Tao. One heuristic justification for the Cohen-Lenstra conjectures, due to Friedman and Washington, relies on the remarkable fact that if M is a random nxn matrix in M_n(Z_p), the distribution of coker(M) among finite abelian p-groups approaches a limit as n goes to infinity; so it makes sense to talk about “the cokernel of a large random matrix” without specifying the size. (There’s a fuller discussion of Friedman-Washington in this old post.) […]